Scaling hypotheses, fractals

To get a first crude estimate of the frequency dependence of the modulus we put forward a simple dynamic scaling argument. It uses d in the same spirit as the exponent z in the dynamic scaling hypothesis in phase transitions according to the theory of Halperin and Hohenberg. Consider the modulus of an entropy-dominated or Brownian system which can be written from dimensional analysis as G kT/V, where V is the volume and kT the thermal energy unit. Here K is a fractal... 

Mandelbrot (24) has suggested a different approach to surface irregularity by using fractal dimensions. A recent proliferation of studies has substantiated the hypothesis of self similarity for a number of natural systems ( 71-73V By this approach, surface irregularity scaling is given by the fractal dimension D, whose range is defined as 2 < D < 3 and which is related to the surface area by the proportionality... 

In the models of the irreversible aggregation, general concepts for the physical processes such as scaling and classes of the universality are widely used [10]. The sense of scaling (scale invariance) consists in abstracting from the details of structure and allocation of simple universal features that are the characteristic for a wide class of systems. Frequently used scaling parameters (indices) are the fractal dimensions. The hypothesis of the universality is closely connected to the hypothesis of scaling whose essence comprises... 

In Figure 5-3, the suscephbility /(< )/( -I-1) is reported. Therefore, from equation (5-7) and from the hypothesis that C(independent measurement of the fractal dimension from SANS experiments, D= 2.4 (Vacher, 1988), and of the spectral dimension by INS experiments, d= 1.3 0.1 (Vacher, 1989, 1990 Courtens, 1990), aUowed the determination of the scaling exponent of C(Different models have been proposed for the calculahon of C(ft>), based on the scaling properties ofthe local strain in fractals (Boukenter,... 

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